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Three consecutive vertices of a parallelogram $\mathrm{ABCD}$ are $\mathrm{A}(6,-2,4) \mathrm{B}(2,4,-8), \mathrm{C}(-2,2,4)$. Find the co-ordinates of the fourth vertex.
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Verified Answer
Let the fourth vertex be $\mathrm{D}(\mathrm{x}, \mathrm{y}, \mathrm{z})$
Mid point of $A C=$ Mid point of $B D$
$$
\begin{aligned}
&\Rightarrow \quad(2,0,4)=\left(\frac{2+x}{2}, \frac{4+y}{2}, \frac{-8+z}{2}\right) \\
&\Rightarrow \quad x=2, y=-4, z=16 \\
&\Rightarrow \quad \text { Forth vertex }=D(2,-4,16)
\end{aligned}
$$
Mid point of $A C=$ Mid point of $B D$
$$
\begin{aligned}
&\Rightarrow \quad(2,0,4)=\left(\frac{2+x}{2}, \frac{4+y}{2}, \frac{-8+z}{2}\right) \\
&\Rightarrow \quad x=2, y=-4, z=16 \\
&\Rightarrow \quad \text { Forth vertex }=D(2,-4,16)
\end{aligned}
$$
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